Ion-ion correlations across and between electrified graphene layersTrinidad Mendez-Morales,1, 2, 3 Mario Burbano,1, 2, 3 Matthieu Haefele,1 Benjamin Rotenberg,2, 3 and MathieuSalanne1, 2, 3, a)1)Maison de la Simulation, CEA, CNRS, Univ. Paris-Sud, UVSQ, Universite Paris-Saclay, 91191 Gif-sur-Yvette,France2)Sorbonne Universite, CNRS, Physico-chimie des electrolytes et nanosystemes interfaciaux, PHENIX,F-75005 Paris, France3)Reseau sur le Stockage Electrochimique de l’Energie (RS2E), FR CNRS 3459,France

When an ionic liquid adsorbs onto a porous electrode, its ionic arrangement is deeply modified due to ascreening of the Coulombic interactions by the metallic surface and by the confinement imposed upon it bythe electrode’s morphology. In particular, ions of the same charge can approach at close contact, leading tothe formation of a superionic state. The impact of an electrified surface placed between two liquid phases ismuch less understood. Here we simulate a full supercapacitor made of the 1-butyl-3-methylimidazolium hex-afluorophosphate and nanoporous graphene electrodes, with varying distances between the graphene sheets.The electrodes are held at constant potential by allowing the carbon charges to fluctuate. Under strongconfinement conditions, we show that ions of the same charge tend to adsorb in front of each other acrossthe graphene plane. These correlations are allowed by the formation of a highly localized image charge onthe carbon atoms between the ions. They are suppressed in larger pores, when the liquid adopts a bilayerstructure between the graphene sheets. These effects are qualitatively similar to the recent templating effectswhich have been reported during the growth of nanocrystals on a graphene substrate.

I. INTRODUCTION

Economic growth is directly coupled with the increaseof energy consumption. In this context, society is fac-ing a double energy challenge in the 21st century: aglobal transition from fossil fuels to renewable energiesmotivated not only by the limit on the supply but alsoby the environmental impacts, and the development ofnew devices with an enhanced ability to store energy.With this in mind, it is not surprising that supercapaci-tors and batteries have attracted considerable attentionfrom research agencies and industry due to their differentproperties and benefits for different applications. Theydiffer in the way they store energy: Whereas superca-pacitors use a reversible adsorption of the ions on theelectrode surfaces with no charge transfer, batteries relyon a reversible redox reaction that occurs in the bulk ofthe electrode. As a consequence, supercapacitors have amuch higher charge and discharge rate and can survivemany more cycles than batteries. Thus, supercapacitorsprovide a much greater power density, but they are stillnot able to store the same amount of energy as the best-performing batteries, such as Li-ion1–3.

In the last years, tremendous research efforts have beendevoted to increasing the energy density of supercapac-itors while maintaining their high power capability andtheir long cycling life4–6. Taking into account that thestored energy in a supercapacitor can be calculated as

E = CV 2

2 (C being the total capacitance of the deviceand V the cell voltage), two approaches can be followedto reach the objective of improving their overall per-formance: either to develop new electrode materials7–10

a)Electronic mail: mathieu.salanne@sorbonne-universite.fr

with increased capacitance or to design novel electrolyteswith wider potential windows11–13 to enhance the oper-ating voltage (and theoretical studies also suggest a syn-ergistic enhancement of the two quantities14,15).

Concerning the electrolytes, they strongly affect theperformance of supercapacitors and many eyes are fixedon increasing their stability under applied voltage. Forthis purpose, room temperature ionic liquids (ILs) can beconsidered as very promising candidates, since these ’de-signer solvents’ are very well-known for exhibiting someunique properties such as high thermal stability, highelectrochemical stability, low vapor pressure and non-toxicity16,17.

When it comes to increasing the capacitance, it hasbeen previously shown, both experimentally and com-putationally, that the interaction between the electrodematerial and the electrolyte plays a fundamental rolein optimizing the adsorption of the ions18–20. For ex-ample, a big step forward was taken with the per-ception of the pore size effect and the ion desolva-tion under confinement made by Chmiola et al.21 whenanalyzing tetraethylammonium tetrafluoroborate (TEA-BF4) in acetonitrile (ACN) using carbide-derived car-bons (CDCs). An anomalous increase of the capacitancewhen the ion size matches the pore size was not only ob-served in traditional electrolytes22, but also employingILs23,24. These findings broke with the traditional viewthat mesoporous electrodes (> 2 nm) were necessary toallow the ions to enter the pores with their solvation shellsintact25, and they drew the attention of the computa-tional research community to gain deeper insight intothe electrolyte environment at electrified surfaces26–29

and under nanoconfinement30–33. Molecular dynamics(MD) simulations employing realistic structural modelsof CDC electrodes demonstrated the better efficiency of

2

nanoporous electrodes over the planar ones34, becausethey avoid the occurrence of overscreening effects andthey allow a denser packing of the ions inside the poresdue to the electronic screening by the pore walls of theions’ Coulombic interactions. This latter effect is knownas “superionic state” and it was first reported by Kondratand Kornyshev35,36. Vatamanu et al.37 also showed a no-ticeable improvement in the capacitive storage when de-signing the electrodes with carbon single-chain segments,due to their high content of atomically rough and curvedsurfaces. Moving to ILs confined in slit-like pores, sev-eral groups38,39 used simulations to predict oscillationsof the capacitance as a function of the pore size. Furtherwork insisted on the importance of pore length40 and onthe necessity to account for the pore size distribution tointerpret the experimental data41.

Based on these findings, many experimental groupshave proposed graphene-based electrode materials toenhance supercapacitor performances42–45. In principle,it should provide the largest accessible surface amongcarbon materials since the ions can approach from bothsides of the graphene plane. However the main difficultiesconsisted in i) avoiding the restacking of the layers whendensifying the material and ii) providing a pathway forthe ions to access the whole material. These difficultieswere overcome by developing new synthesis methods.In particular, a supercapacitor made of the 1-ethyl-3-methylimidazolium bis(trifluoromethane)sulfonimide(EMI-TFSI) electrolyte and holey graphene electrodesyielded a capacitance value of 45 F/g46, while the sameIL inside annealed microwave exfoliated graphene oxide(MEGO) reached 130 F/g47. The performances of thesedevices were further expanded through chemical activa-tion: for example the activated MEGO can store up to200 F/g48. The wide range of experimentally measuredvalues is due to the variability in the structure of thematerials, but it calls for further simulation studiesto rationalize the results. Many works have simulatedslit-pores, which neglects the fact that adjacent layers ofliquid which are located on the two sides of a graphenesheet may interact with each other. With the aim ofclarifying this aspect, here we carry out MD simulationsof perforated nanoporous graphene in the IL 1-butyl-3-methylimidazolium hexafluorophosphate (BMIM-PF6).In particular, we analyze the local structure of the ionsinside the pores and we show that in the case where theIL is highly confined, significant correlations betweentwo adjacent layers of liquid arise. This feature isreminiscent of the “graphene transparency” previouslyreported for the wettability of water49 and the growthof nanocrystals50 on supported graphene, and it mayhave important consequences for charge storage ingraphene-based supercapacitor electrodes.

II. SIMULATION DETAILS

MD simulations of pure BMIM-PF6 confined betweengraphene-based electrodes were carried out within anNVE ensemble at an average temperature of 400 K,which was chosen due to the high viscosity of this IL atroom temperature (109.2 mPa· s at 313.2 K vs 7.1 mPa· sat 413.2 K, both at 1 atm)51. Indeed, it has been previ-ously shown that ion packing and hence, the capacitance,shows little changes with increasing the temperature from298 K to 400 K52,53. Thus, the qualitative conclusionsdrawn in this paper can be extrapolated to systems atlower temperatures. The IL, composed of 4194 ions pairs,is represented by means of a coarse-grained model de-veloped by Roy and Maroncelli54, in which the cationsand anions are respectively characterized by 3 and 1 in-teraction sites. The potential parameters used for thecarbon atoms of the electrodes were σC = 3.37 A andεC = 0.23 kJ/mol following Cole and Klein55. The LJparameters for dissimilar atoms were computed using theLorentz-Berthelot mixing rules.

Each of the electrodes is composed of 21245 carbonatoms distributed into six fixed graphene layers. The fiveinner layers are randomly perforated with round 10 A-radius holes (2 per layer) in order to allow the diffusion ofthe ions in the z direction (perpendicular to the surface)and the entrance of the liquid into the porous electrodeswith fast charging and discharging rates56. With the aimof analyzing the effect of the slit-pore width on the capac-itive properties of the system, two distances between con-secutive graphene layers (7 A and 10 A) were considered.The electrolyte is confined between the two electrodesas shown in Fig. 1. Periodic boundary conditions areapplied along two directions, x and y, and electrostaticinteractions are calculated employing a two-dimensionalEwald summation57.

In order to obtain a realistic description of the prop-erties of these devices, the simulations were carried outby holding the electrodes at constant potential58. Themodel, developed by Reed et al.57 based on the previouswork of Siepmann and Sprik59, uses fluctuating chargeson each carbon atom that are calculated at each timestep by requiring that the potential on every atom is con-stant and equal to the specified electrode potential. Toachieve this polarization of the electrodes, the charges onthe electrode atoms are represented by Gaussian distri-butions centered on the carbons, and they are calculatedby using a conjugate gradient method to minimize theexpression

U =∑

i

qi(t)

[Vi({qj(t)})

2+

qi(t)√2πκ

− V 0

](1)

where qi is the charge on an electrode atom i,Vi({qj(t)}) is the potential felt by an electrode atom atposition i due to the other electrode atoms and all theions in the electrolyte, κ is the width of the Gaussian

3

FIG. 1. Snapshot of the simulation box: BMIM+ cations (blue) and PF−6 anions (red) confined between two graphene-basedelectrodes (green) with a pore size of 7 A. A coarse-grained model is used for the electrolyte.

distributions (κ = 0.5055 A in our case), and V 0 is thepotential applied to the electrode.

The systems were first equilibrated at constant chargeduring 500 ps, where the charge of the electrodes wasfixed at 0. Then we performed a second equilibration of500 ps, in which the charges of the carbon atoms were setto +0.01e and −0.01e in the left and the right electrodes,respectively. We then carried out production runs (witha time step dt = 2 fs) for around 1.3 ns using the con-stant applied potential method. Since this applied po-tential can not be chosen by calculating the Poisson po-tential across the cell for the constant charge runs60, weperformed several short simulations with different start-ing potentials and we chose the one for which the totalcharge on the electrodes showed to be nearly constant:This yielded values of ∆Ψ =1.8 V and 2.7 V for theapplied potential for the 7 A and 10 A pore sizes, respec-tively. All the comparisons are thus performed for almostsimilar electrode charges, which is more meaningful sincethe ionic charge excess is then identical.

III. RESULTS AND DISCUSSION

A. Capacitance

The gravimetric capacitance of the simulated superca-pacitor is given by

Cm =1

m

〈Q+〉∆Ψ

(2)

where 〈Q+〉 is the average charge of the positive electrodealong the full simulation and m is its corresponding mass.However, in order to compare our results to experimen-tal data, it would be necessary to determine single elec-trode capacitances, which is difficult because we cannotcalculate the Poisson potential across the cell for suchcomplex pore geometries60. Nevertheless, our previousestimations in the case of CDC electrodes showed thatfor such a relatively simple IL (in particular with a rel-ative symmetry for the ionic sizes), the two electrodeshad almost similar capacitances. We will therefore as-sume that C+

m ≈ C−m ≈ 2Cm for comparison purposes.We then obtain individual capacitances of 96 and 64 F/gfor the pore sizes of 7 and 10 A, respectively. This fallswell within the range of measured values in non-activatedgraphene-based supercapacitors (i.e. between 45 and 130F/g46,47 as discussed above).

B. Layering inside the pores

In previous studies performed on single slit-pore elec-trodes, an increase of the capacitance was observedwhen decreasing the pore size down to the ionic dimen-

4

-3 -2 -1 0 1 2 30

20

40

60d = 7Å, ∆Ψ = 0.0V

-3 -2 -1 0 1 2 30

5

10

15BMIM+ - ringBMIM+ - methylBMIM+ - butylPF6

-

d = 10Å, ∆Ψ = 0.0V

-3 -2 -1 0 1 2 30

20

40

60d = 7Å, ∆Ψ = 1.8V

-3 -2 -1 0 1 2 30

5

10

15d = 10Å, ∆Ψ = 2.7V

ρ (m

olec

ules

/ nm

3 )

z (Å) z (Å)

ρ (m

olec

ules

/ nm

3 )

FIG. 2. Number density profiles inside the central pore of a neutral (top) and positive (bottom) electrodes, in the directionperpendicular to the electrodes (the z coordinate is centered in the middle of the pore). d is the pore size (7 A in the leftcolumn and 10 A in the right one) and ∆Ψ the applied potential.

sions32,38,39,61. This result was attributed to the tran-sition from a monolayer to a bilayer structure for theadsorbed IL. To check whether the differences in the ca-pacitance we observe between the two pore sizes arisefrom similar effects, we analyzed the structure of theIL inside each of the pores. Number densities (A−3) ofthe ions along the z-direction normal to the electrodeswere calculated in both cases and the profiles obtainedin the neutral and positive electrodes are shown in Fig.2. When the width of the pore is 7 A, we observe thatthe strong confinement of the walls forces the cations andthe anions to occupy a single narrow layer in the centerof the pores of both electrodes (Fig. 2a), with only thesmaller methyl groups of cations approaching closer tothe surface. Since the neutral electrodes (i.e. for an ap-plied potential of 0 V) are already wet by the IL, apply-ing a potential difference leads to a greater intercalationof counter-ions inside the electrodes and the expulsionof some co-ions from the pores, whereas the total num-ber of ions per unit volume of the nanopores remainsnearly the same. In this case, the IL is organized in atight configuration that impedes an increase in ion den-sity upon increasing the applied potential, due to whichthe surface charge accumulation takes place by means ofa swapping mechanism. In this case, even though the co-ions are in direct contact with the charged surface due tothe monolayer confinement, we did not observe a regime

of complete intra-pore co-ion depletion.

On the contrary, in the widest pore (10 A), the distri-bution of the ions is markedly different (Fig. 2b). In thiscase, when the system is not charged the ions accumu-late in two separate layers near the pore walls, since thegreater available space in the pores allows larger fluctu-ations in the structural distribution of the ions. It mustbe noted that the total number of ions adsorbed into theelectrodes (given by the integral of the density) increasesonly slightly with the increasing pore size; which indi-cates that the smaller the pores, the more tightly packedthe ions35. The bilayer structure therefore does not con-sist of two monolayers. Once we apply a potential dif-ference the ion population inside the positive electrodechanges in the same way as noted above (ion exchange)but now, there is also a rearrangement of anions that tendto approach closer to the surface. The minimum betweenthe two anionic density peaks is thus more pronouncedthan in the neutral electrode.

In the negative electrode, the charging mechanism forthe 7 A pores is almost symmetrical with respect to thepositive one. This is not the case for the 10 A pores wherewe observe that almost all the anions concentrate in themiddle of the pores, thus losing their bilayer structure,as shown on Fig. 3. This is reminiscent of the struc-ture which was reported for the same IL adsorbed insideCDC electrodes58, in which the co-ions tend to occupy

5

-3 -2 -1 0 1 2 3z (Å)

0

2

4

6

8

10ρ

(mol

ecul

es/n

m3 )

-1.35V 0.00V+1.35V

FIG. 3. Number density profiles for the anions adsorbed inthe central pore of the 10 A pore size electrode, for appliedpotentials of 0 V (red) and 2.7 V (black: negative electrode,green: positive electrode).

Anion-Cation Cation-Anion

posit. elec. negat. elec.

CDC (1.0 V) 2.6 2.5

d = 7A (0.0 V) 2.9 2.7

d = 7A (1.8 V) 2.3 2.2

d = 10A (2.7 V) 2.3 2.3

TABLE I. Coordination numbers of cations surrounding an-ions in the positive electrode (left), and anions surroundingcations in the negative electrode (right). The results in CDCswere extracted from reference 63. For comparison, the bulkvalues for A-C and C-A are equal to 4.8.

the center of the pores, but also to the case of an ILadsorbed in the interlayer of negatively charged clays62.Again, we observe that the structure of this monolayerof anions differs markedly from the one observed in the7 A pores since the corresponding peak in the density ismuch wider and less high.

C. Decoordination inside the pores

As it was previously reported22, a partial desolvationof the ions occurs under confinement (or rather a deco-ordination in solvent-free ILs). The associated loss ofstability is compensated by the charge of the electrodesand it has been shown to be directly related with themaximum capacitance for pores smaller than the solvatedion size21,64. In Table I we can observe that the coordi-nation numbers of cations around anions (left) and viceversa (right) decreased from 4.8 in the bulk to less than3 inside the nanopores. The coordination numbers ineach electrode were calculated by averaging the numberof co-ions that were closer to a central counter-ion than

FIG. 4. Illustration of the breaking of the Coulombic ordering.Snapshot of an adsorbed layer in the positively charged 7 Apore (blue: imidazolium ring of the cations, red: anions).Pairs (and triplets) of co-ions are surrounded by black ovals.

7.5 A, which is the first minimum of the anion-cationradial distribution function (RDF) in the bulk. Whencompared with CDCs63, the ions confined in perforatedgraphene-based electrodes are, on average, less coordi-nated. It must be noted that although the distributionsof the coordination numbers are centered around 2 inboth cases, in the CDC case occurrences of up to 7 areobserved, which is due to the presence of different ad-sorption sites63. As expected, a greater desolvation of thecounter-ions is obtained inside charged electrodes. How-ever, we did not register an influence of the pore size onthe coordination numbers, because even though the iondensity is greater in the 7 A-system, the 2D-solvationof the ions is characterized by larger lateral distancesbetween the co-ions remaining inside the pores and thecounter-ions introduced into them51.

D. Partial breaking of the Coulombic ordering

Our results suggest the stabilization of the partially de-solvated counter-ions inside the nanopores by means ofa charge compensation from the electrodes, which couldbe explained by the picture reported by Kondrat andKornyshev35,36 of an exponential screening of the electro-static interactions in a slit-like metallic pore due to theimage charges. This superionic state breaks the Coulom-bic ordering characteristic of ILs and allows more ionsof the same sign to occupy the pore, which leads to a

6

high concentration (mainly limited by steric repulsions)of partially desolvated counter-ions that can be in con-tact with the charged walls. The screening of the elec-trostatic interactions is expected to sharply increase innarrower pores, thus yielding a significant improvementof the capacitive properties. This theoretical finding hasalso recently been confirmed experimentally by perform-ing X-ray scattering experiments to resolve the structureof the EMI-TFSI ionic liquid confined inside electrifiednanoporous carbons65. To analyze the presence of such adense ionic state in our simulation, we computed the ra-tio of the number of anion-anion (cation-cation) pairs inthe positive (negative) electrode to the number of anions(cations) in that electrode. We considered the formationof a pair of ions with similar charges if they are locatedcloser than 5.8 A, which is the distance at which theanion-anion (or cation-cation) and anion-cation RDFs(both in the bulk) intersect each other (for cations theinteraction center corresponding to the imidazolium ringwas used to perform this analysis). A set of such pairs ishighlighted on a snapshot of the central adsorbed layerin the positively charged 7 A pore on Figure 4. We canobserve that the fractions increase with the confinement,switching from 0.049 to 0.109 for anions in the positiveelectrode and from 0.099 to 0.141 for cations in the nega-tive one. Albeit on a different system, these numbers arein good agreement with the ones reported in the experi-mental study of Futamura et al.65, thus providing furtherevidence for the existence of the superionic state for ionicliquids adsorbed inside ultranarrow electrified nanopores.

E. Ion-ion correlations from pore to pore

The results described above concerning the formationof mono/bilayer(s), decoordination and the formation ofpairs of anions or cations are in agreement with the pre-vious literature on single slit pore simulations. However,our original simulation setup allows the study ionic in-teractions from one pore to another. In a recent work,a templating effect of graphene was recently reported byChae et al.50. They showed that the atomic arrange-ments of ZnO nanocrystals nucleated on a graphene planeexhibit a close match with those of a substrates bound tothe other side of the graphene layer. In our case a sim-ple visualization of the atomic positions in two adjacentpores (Figure 5a) suggests that similar mechanisms maybe at play. In order to investigate the existence of suchcorrelations between ions adsorbed in adjacent pores, wecompute the two-dimensional radial distribution function(RDF) between an ion within a given interlayer and theions in a neighbouring one. As illustrated on Figure 5band c, we project an anion I with coordinates (xI , yI , zI)from the layer i to the layer j. The coordinates of thisprojection are then (xI , yI , zI + d) where d is the poresize. We then define an in-plane distance between thisprojected anions and any anion J present inside layer jas

r′IJ =√

(xI − xJ)2 + (yI − yJ)2 (3)

We then perform an ensemble average to determine thein-plane anion-anion RDF between layers i and j (gij):

gij(r′) =

S

NINJ2πr′dr′〈∑

I∈i

∑

J∈jδ(r′ − r′IJ)〉 (4)

where NI and NJ are the number of anions inside lay-ers i and j, respectively, S is the lateral surface of thesimulation cell (along x and y coordinates) and dr′ isthe precision chosen for binning the Kronecker functionδ. Similar RDFs can be defined for cation-cation andanion-cation pairs.

Examples of such functions are provided for varioussimulation setups in Figure 6 for the correlations be-tween the fourth and fifth layers of liquid (similar re-sults were obtained for all the adjacent layers). Firstly,very large differences are observed between the electrifiedpores with a size of 7 A (top) and 10 A (middle). Indeed,for the former we observe a strong peak at r = 0 A for theanion-anion, which shows that anions have a preferencefor adsorbing in front of other anions. On the contrary,the cation population is depleted in such sites as can beseen from the low value taken by the anion-cation RDF.The cation-cation RDF is qualitatively similar, albeit itsshape is smoother due to the greater flexibility that isallowed to the BMIM+ in the coarse-grained model. Thestructure of each fluid layer is therefore clearly corre-lated with that in the neighbouring ones across graphenesheets, with fluctuations due to the temperature and thepresence of holes inside the graphene plane.

In the 10 A pores, these correlations are almost com-pletely lost. A small preference for like-charge ions is stillobserved since the anion-anion and cation-cation RDFsslightly rise above 1 for short distances, but this effect isalmost not noticeable. It is therefore necessary to have awell-defined single layer of ions to observe the effect.

In order to check whether these correlations arise fromthe image charges induced by the ions themselves on thesurface of the carbon, we performed an additional simu-lation for the 7 A pores where the average charge of theplane was equally shared between all the carbon atoms.This situation is often called “constant charge” simula-tion, although it does not correspond to a case relevant toelectrochemical simulations. As shown in the correspond-ing RDFs (bottom panel of Figure 6), the correlations arethen lost. On the contrary, Coulombic ordering is recov-ered and cations have a slight tendency to lie in front ofanions from adjacent layers. This shows that the chargelocalization on the graphene surface plays a key role andthat constant charge simulations should be avoided whensimulating complex porous carbons in which the surfaceis accessible from the two sides.

7

Layer i Layer j

Ion I

b)a)

z x

y

6

Layer i Layer j

Ion Ib)

0 2 4 6 8 100

1

2

3

0 2 4 6 8 10r (Å)

0

1

2

3 Anion - AnionCation - CationCation - Anion

d=7 Å, ∆Ψ=1.8 V

d=10 Å, ∆Ψ=2.7 V

g'45

(r)

c)

52

Introduc*on

MD

simula*ons

Results

Modelling nanoporous graphene-based supercapacitors

• CORRELATION BETWEEN LAYERS:

a)

FIG. 5. In plane radial distribution functions between an ion within the third interlayer and the ions in the fourth one. Thedistance r0 corresponds to the relative distance along the surface. Top: A constant potential is applied to the electrode,allowing the surface charge to fluctuate in response to the electrolyte. Bottom: The partial charges on the carbon atoms areheld constant.

r0 =p

x2 + y2 (3)

An example of such a function is provided in the tippanel of 5 for an anion in the third interlayer and theanions and cations inside the fourth one. The peak at r= 0 A shows that anions have a preference for adsorbingin front of other anions. On the contrary, the cationpopulation is depleted in such sites. The structure ofeach fluid layer is therefore clearly correlated with thatin the neighbouring ones across graphene sheets, withfluctuations due to the temperature and the presence ofholes inside the graphene planes.

In order to check whether these correlations arise fromthe image charges induced by the ions themselves on thesurface of the carbon, we performed an additional simu-lation where the average charge of the plane was equallyshared between all the carbon atoms (a situation oftencalled ”constant charge” simulation, although it does notcorrespond to a case relevant to electrochemical simula-tions). As shown in the corresponding radial distributionfunctions (bottom panel), the correlations are then lost.On the contrary, Coulombic ordering is recovered andcations have a slight tendency to lie in front of anionsfrom adjacent planes.

IV. CONCLUSIONS

In summary, BMIM-PF6 structure in nanoporousgraphene-based electrodes was analyzed by means of MDsimulations that take into account the polarizability ofthe walls. Both anions and cations were significantly de-solvated inside the nanopores regardless their width, butthe way they are accommodated varied from a single nar-row layer to a bilayer configuration when increasing the

pore size from 7 A to 10 A. Additionally, the ions werefound to be more densely packed in the smaller pores,which is possible due to the screening of the Coulombicinteractions between the ions by the image charges. This’superionic state’ previously predicted by Kondrat andKornyshev was confirmed by a more marked formationof counter-ion/counter-ion pairs under a strong confine-ment. We also considered the relation between the inte-gral capacitance and the size of the pores. However, ourresults did not correspond with the experimental achieve-ments and they did not show any dependence on the porewidth. This unexpected behaviour of the capacitance wasexplained in terms of the correlation between ions of thesame sign belonging to di↵erent pores; that is, the dis-tribution of the ions in one layer is strongly influencedby the conformation that the ions adopt inside the otherlayers due to the image forces. This correlation, whichshowed to be much more remarkable in the thinner pores,leads to a picture in which the adsorption of the ions and,consequently, the performance of the supercapacitor, islimited by the less e�cient pores.

From these evidences we can conclude that tuning thepore size to fit the size of the ions of the electrolytewith the aim of enhancing the capacitance of nanoporouscarbon-based supercapacitors is not enough. Our re-sults point to the morphology of the electrodes havinga deep influence on the charge storage and show thatstructural disorder, surface roughness and pore curva-ture are key factors when it concerns the design of newhigh-performance supercapacitors.

c)

FIG. 5. a) Snapshot of a a portion of the system representing two adjacent pores separated by a graphene sheet for thesimulation performed with d = 7A and ∆Ψ =1.8 V (red: anions, blue: imidazolium rings of the cations, green: carbon atoms).b) Projection of the ion I from layer i to the layer j. c) The in-plane distances r′ between the projected ion and all the ions inlayer j are then computed. Note that panels b) and c) show all the atoms in the PF−6 anion, but a coarse-grained model wasused in the simulation.

IV. CONCLUSIONS

In summary, BMIM-PF6 structure in nanoporousgraphene-based electrodes was analyzed by means of MDsimulations using the constant applied potential methodto simulate the electrode materials. Both anions andcations were significantly desolvated inside the nanoporesregardless of their width, but the way they are accommo-dated varied from a single narrow layer to a bilayer con-figuration when increasing the pore size from 7 A to 10 A.Additionally, the ions were found to be more denselypacked in the smaller pores, forming a single monolayerwhile they arrange in two layers when they have accessto a larger volume. The charging of the supercapacitoris then allowed by an efficient screening of the Coulom-bic interactions between the ions by the image charges.This superionic state previously predicted by Kondratand Kornyshev was confirmed by a more marked for-mation of counter-ion/counter-ion pairs under a strongconfinement.

The main specificity of the presence of a singlegraphene sheet between adjacent layers of liquids is thatit allows the presence of structural correlations betweenions of the same sign belonging to different pores; thatis, the distribution of the ions in one layer is strongly in-fluenced by the conformation that the ions adopt insidethe other layers due to the image forces. It is worthnoting that in a former DFT study, it was shown thattwo BF−4 ions adsorbing on the same site on both sidesof a graphene layer were more stable than those on dif-ferent sites66. Here the correlation effects seem to beenhanced by a strong confinement since they are muchmore remarkable in the thinner pores, and almost ab-

sent when a bilayer of liquid is allowed to form inside theelectrode. We proved that here again, the strong local-ization of image charges on the graphene surface plays akey role: When assigning similar constant charges to thecarbon atoms (with the same average as in the constantpotential simulation), the correlations are reversed andthe usual Coulombic ordering is recovered. These resultsopen further questions for the development of efficientgraphene-based supercapacitors. For example, it will beof great interest to determine whether these correlationsremain present in realistic models, where restacking ofthe graphene layers is likely to happen. Much of the ap-plications involve organic electrolytes67 or solvate ionicliquids68, so that it will also be of importance to deter-mine if the presence of a solvent mitigates the importanceof the ion-ion correlations across the graphene layers.

ACKNOWLEDGMENTS

This work was supported by the French National Re-search Agency (Labex STORE-EX, Grant No. ANR-10-LABX-0076 and ANR SELFIE, Grant No. ANR-17-ERC2-0028) and by Defi CNRS INPHYNITI 2015-2016 (SIMELEC). We acknowledge support from Eo-CoE, a project funded by the European Union ContractNo. H2020-EINFRA-2015-1-676629, from the DSM-nergie programme of CEA and from the Eurotalent pro-gramme. We are grateful for the computing resources onOCCIGEN (CINES, French National HPC) and CURIE(TGCC, French National HPC) obtained through theproject x2016096728.

8

0 2 4 6 8 100

1

2

3g 45

(r')

0 2 4 6 8 100

1

2

3

g 45(r'

)

0 2 4 6 8 10r' (Å)

0

1

2

3

g 45(r'

)d=7 Å, ∆Ψ=1.8 V

d=10 Å, ∆Ψ=2.7 V

- -

- -

d=7 Å, constant charge

- -

+ +

+ +

+ +

+

+

+-

-

-

a)

b)

c)

FIG. 6. In-plane radial distribution functions between an ionwithin the fourth layer of adsorbed liquid and the ions in thefifth one (black: anion-anion, red: cation-cation and green:anion-cation). Panels a) and b): A constant potential is ap-plied to the electrode, allowing the surface charge to fluctu-ate in response to the electrolyte (a: d = 7A ∆Ψ=1.8 V,b: d = 10A ∆Ψ=2.7 V). Panel c): The partial charges onthe carbon atoms are held constant and equal to the averagecharge for the 1.8 V applied potential simulation (d = 7A).

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